“This court has recognized that they have a high rate of recidivism and are very likely to do this again,” said the lawyer, Robert C. Montgomery, who was defending a North Carolina statute that bars sex offenders from using Facebook, Twitter and other social media services.

The Supreme Court has indeed said the risk that sex offenders will commit new crimes is “frightening and high.” That phrase, in a 2003 decision upholding Alaska’s sex offender registration law, has been exceptionally influential. It has appeared in more than 100 lower-court opinions, and it has helped justify laws that effectively banish registered sex offenders from many aspects of everyday life.

But there is vanishingly little evidence for the Supreme Court’s assertion that convicted sex offenders commit new offenses at very high rates. The story behind the notion, it turns out, starts with a throwaway line in a glossy magazine.

Justice Anthony M. Kennedy’s majority opinion in the 2003 case, Smith v. Doe, cited one of his own earlier opinions for support, and that opinion did include a startling statistic. “The rate of recidivism of untreated offenders has been estimated to be as high as 80 percent,” Justice Kennedy wrote in the earlier case, McKune v. Lile.

The guide, a compendium of papers from outside experts, is 231 pages long, and it contains lots of statistics on sex offender recidivism rates. Many of them were in the single digits, some a little higher. Only one source claimed an 80 percent rate, and the guide itself said that number might be exaggerated.

The source of the 80 percent figure was a 1986 article in Psychology Today, a magazine written for a general audience. The article was about a counseling program run by the authors, and they made a statement that could be good for business. “Most untreated sex offenders released from prison go on to commit more offenses — indeed, as many as 80 percent do,” the article said, without evidence or elaboration.

That’s it. The basis for much of American jurisprudence and legislation about sex offenders was rooted in an offhand and unsupported statement in a mass-market magazine, not a peer-reviewed journal.

“Unfortunately,” Melissa Hamilton wrote in a new article in The Boston College Law Review, “the Supreme Court’s scientifically dubious guidance on the actual risk of recidivism that sex offenders pose has been unquestionably repeated by almost all other lower courts that have upheld the public safety need for targeted sex offender restrictions.”

The most detailed examination of how all of this came to pass was in a 2015 article in Constitutional Commentary by Ira Mark Ellman and Tara Ellman, who were harshly critical of the Supreme Court.

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“Its endorsement has transformed random opinions by self-interested nonexperts into definitive studies offered to justify law and policy, while real studies by real scientists go unnoticed,” the authors wrote. “The court’s casual approach to the facts of sex offender re-offense rates is far more frightening than the rates themselves.”

There are many ways to calculate recidivism rates, and they vary depending on a host of distinctions. A 2014 Justice Department report found, for instance, that sex offenders generally have low overall recidivism rates for crimes. But they are more likely to commit additional sex offenses than other criminals.

In the three years after release from prison, 1.3 percent of people convicted of other kinds of crimes were arrested for sex offenses, compared to 5.3 percent of sex offenders. Those findings are broadly consistent with seven reports in various states, which found that people convicted of sex crimes committed new sex offenses at rates of 1.7 percent to 5.7 percent in time periods ranging from three to 10 years.

The Justice Department report said the risk of new sex offenses by convicted sex offenders rises over time, reaching 27 percent over 20 years.

That number is significant, but it is nothing like 80 percent. Perhaps it is sufficient to warrant harsh sex offender registry laws, but judges and lawmakers would have been better served by basing their judgments on the best available data.

Lower courts generally accept what the Supreme Court says. That is true not only about the law but also about facts subject to independent verification. Last year, though, the federal appeals court in Cincinnati gently suggested that the Supreme Court had taken a wrong turn in its 2003 decision in Smith v. Doe.

Judge Alice M. Batchelder, writing for a unanimous three-judge panel, described “the significant doubt cast by recent empirical studies on the pronouncement in Smith that ‘the risk of recidivism posed by sex offenders is “frightening and high.’”

The appeals court struck down a particularly strict Michigan sex-offender law as a violation of the Constitution’s ex post facto clause, saying it retroactively imposed punishment on people who had committed offenses before the law was enacted. The state has asked the Supreme Court to consider the case, Does v. Snyder, No. 16-768. The first paragraph of its petition says that the risk of recidivism “remains ‘frightening and high.’”

The constitutional question in the case is interesting and substantial. And hearing the case would allow the court to consider more fully its casual assertion that sex offenders are especially dangerous.

A version of this article appears in print on March 7, 2017, on Page A10 of the New York edition with the headline: Not a Lot of Evidence Behind a Supreme Court Ruling on Sex Offenders. Order Reprints|Today's Paper|Subscribe